Advances in Quantum Monte CarloâPast, Present, and Reflections on

Chapter 11

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Stuart M. Rothstein* and Egor Ospadov Department of Physics, Brock University, St. Catharines, Ontario L2S 3A1, Canada *E-mail: [email protected].

In his introduction to the year 2000 symposium volume, M.H. Kalos summarized themes presented at that meeting, with his thoughts on future challenges facing quantum Monte Carlo researchers. Similarly, in the year 2015 symposium volume, J.B. Anderson surveyed progress made in the field up to that point and future directions. Our contribution continues in a similar vein. We classify by major theme papers presented at the 2015 meeting and relate them to those published in earlier proceedings volumes with a similar focus. We conclude with reflections on current challenges in the field and speculate on new major themes for papers that we anticipate to be presented at future meetings.

Introduction This paper is organized as follows: first, we allude to previous overviews of the field that appeared in the years 2000 and 2010 symposium volumes. Therein notable experts had identified several challenges facing quantum Monte Carlo researchers at those points in time. Next, to demonstrate how the field has evolved to meet these, we relate papers presented at the 2015 meeting to those with the same broad themes appearing in earlier symposium volumes. And finally, we reflect on challenges presented by chemical applications that remain to be addressed in these symposia, and we speculate on major new themes that may be developed in forthcoming meetings.

© 2016 American Chemical Society Tanaka et al.; Recent Progress in Quantum Monte Carlo ACS Symposium Series; American Chemical Society: Washington, DC, 2016.

Downloaded by COLUMBIA UNIV on December 2, 2016 | http://pubs.acs.org Publication Date (Web): December 1, 2016 | doi: 10.1021/bk-2016-1234.ch011

Below, we make frequent references to papers that appear in monographs associated with the Advances in Quantum Monte Carlo meetings held from 1995 to 2010 (1–4). We employ a more-or-less obvious notation like that used for the first entry in Table 1, which reads: D. Bressanini et al. 3 2002 (2), where 3 represents the page number, 2002 identifies the publication year, and (2) specifies the reference cited in this work. This is equivalent to: Bressanini, D.; Reynolds, P. J. in Recent Advances in Quantum Monte Carlo Methods, Part II; Lester Jr., W. A., Rothstein, S. M., Tanaka S., Eds.; Recent Advances in Computational Chemistry; World Scientific, Singapore. 2002; Vol 2; p. 3. In a similar vein, a citation like D. M. Ceperley et al. 2015 refers to the paper presented by the indicated authors at the 2015 symposium.

Previous Overviews of the Field of Quantum Monte Carlo In the 2002 Symposium volume, together with a summary of major themes presented at the year 2000 meeting, M.H. Kalos identified a number of challenges met by researchers in the field and those remaining to be resolved. He identified the following: “the ‘Fermion sign problem’; the issue of small perturbations; the question of estimating the square of the wave function without using ‘extrapolation’ or some variant of ‘forward walking’; the problem of large-Z slowing down; the treatment of excited states; and a natural non-perturbative method for relativistic effects”. Furthermore, he recognized that although the conference papers rightly focused on applications to chemistry, quantum Monte Carlo has enjoyed wide successes in several other fields: particle physics, nuclear physics, solid state physics and low temperature physics (5). Ten years later, in these volumes, J.B. Anderson gave an extensive review of significant developments in quantum Monte Carlo methods, virtually from its first applications to chemistry, under the following broad headings: small molecules, the electron gas, pseudopotentials, stiff and floppy molecules, van der Waals systems and solvated molecules. Although he didn’t dwell on challenges to quantum Monte Carlo researchers per se, he clearly identified stiff and floppy molecules and perhaps all polyatomic molecules as presenting difficulties in quantum chemistry. At the time of writing his review, 2012, Anderson cited brand new work on quantum mechanical treatments of solvated molecules, in anticipation of further successes in that endeavor (6).

Advances in Quantum Monte Carlo – Past and Present Towards Meeting the Challenges As evidenced by contributions to earlier symposium volumes and to papers presented at the 2015 meeting, this section illustrates how research in quantum Monte Carlo has evolved to meet the methodological challenges posed above by Kalos and Anderson (Tables 1-5). 156 Tanaka et al.; Recent Progress in Quantum Monte Carlo ACS Symposium Series; American Chemical Society: Washington, DC, 2016.


Table 1. Fermion Sign Problem, Nodes, and Exact Treatments Title

Author(s)

Page

Year (ref)

Mathematically determined nodes

D. Bresanini et al.

3

2002 (2)

Variety of auxiliary field QMC

Y. Asai

40

2002 (2)

Shifted contour auxiliary field QMC

R. Baer et al.

279

2002 (2)

Fixed hypernode method

F. Pederviva et al.

81

2007 (3)

Fermion Monte Carlo

M. Kalos et al.

93

2007 (3)

Exact treatment of fermions

N. H. Tubman et al.

41

2012 (4)

Many-body nodal hypersurces

S. Hu et al.

77

2012 (4)

Exact quantum Monte Carlo projection methods

D. M. Ceperley et al.

2015

Tensor network wave functions

G. K.-L. Chan et al.

2015

Brueckner-Glodstone QMC

S. Hirata

2015

Compressed sensing

A. Aspuru-Gizik

2015

Table 2. Small Perturbations Title

Author(s)

Page

Year (ref)

Difference between true and trial wave functions

J. B. Anderson et al.

21

1997 (1)

Correlated sampling in QMC

C. Filippi et al.

12

2002 (2)

Energy differences in DMC

J. B. Anderson

3

2007 (3)

Molecular crystal polymorphism

M. A. Watson et al.

101

2012 (4)

QMC and spin-orbit interation

A. Abrosetti et al.

119

2012 (4)

Positron binding to molecules

Y. Kita et al.

157

2012 (4)

Variable spins

L. Mitas et al.

2015

157 Tanaka et al.; Recent Progress in Quantum Monte Carlo ACS Symposium Series; American Chemical Society: Washington, DC, 2016.


Table 3. Estimating the Square of the Wave Function without Using Extrapolation or Some Variant of Forward-Walking Page

Title

Author(s)

Year (ref)

Pure-sampling QMC

S. M. Rothstein

2015

Molecular Dynamics QMC

S. Sorella

2015

Table 4. Multireference Approaches and Excited Electronic States (in Part: Problem of Large Z Slowing Down) Title

Author(s)

Page

Year (ref)

Hybrid Nonadiabatic QMC

D. Bressannini et al.

65

1997 (1)

Multireference trial functions

H.-J. Flad et al.

73

1997 (1)

Model potential and Positron binding

K. Iguchi

99

1997 (1)

Psuedopotentials

C. W. Greef et al.

117

1997 (1)

Imaginary time spectral evolution

P. Huang et al.

111

2002 (2)

Excited states

R. J. Needs et al.

143

2002 (2)

Rydberg states

A. Bande et al.

43

2007 (3)

Chromium dimer

K. Hongo

91

2012 (4)

Perturbative-selected CI expansions

M. Caffarel

2015

HOMO/LUMO gap

K. Jordan

2015

CI with pairwise MO correlations

S. Tanaka

2015

Antisymmetric geminals

E. Neuscamman

2015

Model space QMC

S. Ten-no

2015

Full CI QMC

A. Alavi

2015

Large CI expansions

A. Scemama

2015



Table 5. Polyatomic Molecules, van der Waals Systems and Solvated Molecules Title

Author(s)

Page

Year (ref)

QMC study of Si and C molecular systems

L. Mitas et al.

133

1997 (1)

Vibratonal properties and quantum dynamics of molecules

S. Tanaka

95

2002 (2)

Hg clusters

H.-J. Flad et al.

183

2002 (2)

Hg clusters

M. C. Wilson et al.

1

2007 (3)

Various loosely-bound complexes

M. J. T. Jordan et al.

101

2007 (3)

Biomolecular calculations using QMC combinded with FMO method

R. Maezono et al.

141

2007 (3)

Vibrational excited states

A. B. McCoy

147

2007 (3)

Properties of floppy molecules

A. B. McCoy

145

2012 (4)

MD/MC applied to liquid He

S. Miura

177

2012 (4)

Water trimer and olgopeptide

T. Fujita et al.

187

2012 (4)

Beyond a single solvated electron

D. Y. Zubarev et al.

201

2012 (4)

QMC in curved space

E. Curotto

2015

Transition metal organometallic clusters

I. Stich

2015

Various loosely-bound complexes

M. Dubecky

2015

Liquid, ice and bulk water clusters

D. Alfe

2015

Rotation/vibration states

A. B. McCoy

2015

Other Significant Advances Quantum Monte Carlo researchers have addressed even more methodological and conceptual challenges that rest at the frontiers of theoretical chemistry (Tables 6-8). Notwithstanding the Advances in Quantum Monte Carlo symposia being hosted within an international chemistry congress, the field’s successes when applied in other areas of physics continues to be evident in symposium monographs and by papers presented at the most recent symposium. (Table 9). 159 Tanaka et al.; Recent Progress in Quantum Monte Carlo ACS Symposium Series; American Chemical Society: Washington, DC, 2016.


Table 6. Optimization and Other Algorithmic issues Title

Author(s)

Page

Year (ref)

Monte Carlo optimization

M. P. Nightingale et al.

201

1997 (1)

Improved scaling with LMOs

A. Luechow et al.

30

2002 (2)

Optimization applied to vdw clusters

M. P. Nightingale et al.

127

2002 (2)

Linear scaling of local energy

B. Austin et al.

55

2007 (3)

Population control bias

J. T. Krogel

13

2012 (4)

Auxiliary potential energy surface

A. Nakayama et al.

27

2012 (4)

QMC and HF symmetry dilemma

P. Reinhardt et al.

53

2012 (4)

Optimzation and other issues

J. Toulouse

2015

Fixed-node projector MC energy

C. J. Umrigar

2015

Table 7. Electronic Properties Title

Author(s)

Page

Year (ref)

Analytical wavfunctions

D. Bressannini et al.

1

1997 (1)

Beryllium atom revisited

S. A. Alexander et al.

55

2002 (2)

Static electrical properties

M. Hornick

71

2002 (2)

Harmonic frequencies

S.-I. Lu

29

2007 (3)

Atomic forces

M. W. Lee et al.

69

2007 (3)

Table 8. Conceptual Challenges Title

Author(s)

Page

Year (ref)

Single electron densities from QMC

A. Luechow

65

2012 (4)

Insight on chemical bond

N. Tubman

2015

Chemical bonding

A. Luechow

2015



Table 9. Quantum Monte Carlo in Other Areas of Physics Title

Author(s)

Page

Year (ref)

Postros: Challenge and opportunities for quantum Monte Carlo

D. M. Schrader

163

1997 (1)

QMC for realistic and model solids

P. Gori-Giorgi et al.

205

2002 (2)

Coupled electronic-ionic Monte Carlo

M. Dewing et al.

218

2002 (2)

Quantum Chemodynamics

M. Nagaoka

254

2002 (2)

Doped He clusters

N. Blinov

165

2007 (3)

High-energy electron scattering

S. A. Alexander

131

2012 (4)

Entanglement entropy in solids

R. G. Melko

2015

Entanglement in quantum fluids and gases

A. Del Maestro

2015

Renyi entanglement entropies

C. M. Herdman

2015

QMC simulations of confined rotors

P.-N. Roy

2015

QMC simulations of doped He clusters

M. Lewerenz

2015

QMC simulations of doped He clusters

R. Hinde

2015

Strongly correlated materials

L. Wagner

2015

Reactions on metal surfaces

P. Hoggan

2015

Advances in Quantum Monte Carlo – Reflections on Its Future Theory and Application A non-perturbative treatment of relativistic effects is the sole, unmetchallenge to quantum Monte Carlo researchers, identified by M. Kalos in his introduction to the year 2002 symposium volume. Will it be met in the 2020 or some future symposium? This remains to be seen, but we anticipate that a number of other major, and arguably less-difficult themes will surface at future meetings: hybrid QM/MM methods; systematic improvement of density functionals; non-trivial electronic properties, such as dynamic polarizabilites; linear scaling and other innovative algorithms; non-adiabatic QMC; novel pseudopotentials, effective potentials, and model potentials; chemical accuracy for “large” molecules; cold chemistry; quantum biology; and chemical reaction mechanisms. 161 Tanaka et al.; Recent Progress in Quantum Monte Carlo ACS Symposium Series; American Chemical Society: Washington, DC, 2016.

Future advances in our field will be greatly facilitated by new computing hardware, albeit with significant changes in the way that quantum Monte Carlo researchers conduct their research. We conclude by discussing this in some detail.


Exascale Computing High-performance computing (HPC) went through immense growth and excitement in the last decade. Roadrunner, a supercomputer commissioned by U.S. Department of Energy and built by IBM, became the first supercomputer to reach the petaflops mark (1015 floating-point operations per second) in June of 2008, ushering the era of petascale computing into reality (7). As of today, there are 82 supercomputers capable of performing operations in excess of one petaflops. Four of these systems are able to perform at over ten petaflops (8). Ever since petascale computing was established, individuals, institutions and governments set their sight towards a new goal: exascale computing, an ability to perform in excess of one exaflops (1018 floating-point operations per second). In Japan, RIKEN is planning to develop an exascale supercomputer by 2020, as a part of their Exascale Supercomputer Project and successor to their K supercomputer. Furthermore, President Obama signed an Executive Order in July 2015, creating a National Strategic Computing Initiative (NSCI) to provide funding for and accelerate the development of exascale computing systems. Currently, the top two supercomputers are designed using hybrid architecture: central processing units (CPUs) are paired up with either graphics processing units (GPUs) or coprocessors. Titan, which holds the second place, is built using AMD Opteron CPUs paired with NVIDIA K20x GPUs (8). Tianhe-2, which holds the first place, is built using Intel Xeon CPUs paired with Intel Xeon Phi coprocessors (8). The use of hybrid architecture is lucrative because it provides a large increase in processing power by allowing the CPU to offload computationally-intensive tasks to the GPU or the coprocessor. In order to take advantage of the growing petascale and future exascale infrastructure, quantum Monte Carlo (QMC) developers need to become computer scientists, because there is no simple way to compile current QMC codes to utilize these systems efficiently. To optimize our programs, we must learn new programming languages and techniques and develop an intricate knowledge of new hardware. We need to understand different memory models employed by different GPUs and coprocessors and the advantages and disadvantages of using CUDA or OpenCL. For example, compiling existing or writing new code to utilize coprocessors is easier than targeting GPUs but in many cases GPUs may be preferred due to their superior efficiency. On the other hand, many GPUs are very slow at executing double precision arithmetic. A few groups already started to utilize GPUs in their QMC calculations (9–12) and reported a substantial performance increase of up to 150 times (12), when compared to conventional CPUs. In contrast, the field of molecular dynamics started to experiment with GPUs much earlier and achieved performance increase in excess of 700 times (13). Furthermore, many of their popular packages now support GPUs out of the box: AMBER, Firefly, GROMACS, VMD and NAMD. Notwithstanding the challenges to make it 162 Tanaka et al.; Recent Progress in Quantum Monte Carlo ACS Symposium Series; American Chemical Society: Washington, DC, 2016.

possible, it is evident that conversions of existing quantum Monte Carlo packages to exploit exascale infrastructure are feasible, e.g. (11), and will be even more valuable tools for quantum Monte Carlo researchers.

Acknowledgments We acknowledge useful comments on these topics made by the following individuals: A. Alavi, D. Ceperley, E. Curotto, P. E. Hoggan and S. Tanaka.


References Recent Advances in Quantum Monte Carlo Methods; Lester, W. A., Jr., Ed.; Recent Advances in Computational Chemistry; World Scientific: Singapore, 1997; Vol 2. 2. Recent Advances in Quantum Monte Carlo Methods, Part II; Lester, W. A., Jr., Rothstein, S. M., Tanaka S., Eds.; Recent Advances in Computational Chemistry; World Scientific: Singapore, 2002; Vol 2. 3. Advances in Quantum Monte Carlo; Anderson, J. B., Rothstein, S. M., Eds; ACS Symposium Series No. 953; American Chemical Society: Washington, DC, 2007. 4. Advances in Quantum Monte Carlo; Tanaka, S., Rothstein, S. M., Lester, W. A., Jr., Eds; ACS Symposium Series No. 1094; American Chemical Society: Washington, DC, 2012. 5. Kalos, M. H. In Recent Advances in Quantum Monte Carlo Methods, Part II; Lester Jr., W. A., Rothstein, S. M., Tanaka S., Eds.; Recent Advances in Computational Chemistry; World Scientific: Singapore, 2002; Vol 2; pp ix−x. 6. Anderson, J. B. In Advances in Quantum Monte Carlo; Tanaka, S., Rothstein, S. M., Lester, W. A., Jr., Eds; ACS Symposium Series No. 1094; American Chemical Society: Washington, DC, 2012; pp 209−219. 7. “Roadrunner: Los Alamos National Laboratory.” TOP500; http://www.top500.org/resources/top-systems/roadrunner-los-alamos-nationallaboratory/ (accessed January 22, 2015). 8. “November 2015.” TOP500; http://www.top500.org/lists/2015/11/ (accessed January 22, 2015). 9. Anderson, A. G.; Goddard, W. A.; Schröder, P. Quantum Monte Carlo on graphical processing units. Comput. Phys. Commun. 2007, 177, 298–306. 10. Uejima, Y.; Terashima, T.; Maezono, R. Acceleration of a QM/MM-QMC simulation using GPU. J. Comput. Chem. 2011, 32, 2264–2272. 11. Esler, K. P.; Kim, J.; Ceperley, D. M.; Shulenburger, L. Accelerating Quantum Monte Carlo Simulations of Real Materials on GPU Clusters. Comput. Sci. Eng. 2012, 14, 40–51. 12. Lutsyshyn, Y. Fast quantum Monte Carlo on a GPU. Comput. Phys. Commun. 2015, 187, 162–174. 13. Friedrichs, M. S.; Eastman, P.; Vaidyanathan, V.; Houston, M.; Legrand, S.; Beberg, A. L.; Ensign, D. L.; Bruns, C. M.; Pande, V. S. Accelerating molecular dynamic simulation on graphics processing units. J. Comput. Chem. 2009, 30, 864–872. 1.


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